A Novel Scheme of Strain-constructed Point Interpolation Method for Static and Dynamic Mechanics Problems

This paper presents a new scheme of strain-constructed point interpolation method (SC-PIM) for static, free and forced vibration analysis of solids and structures using triangular cells. In the present scheme, displacement fields are assumed using shape functions created via the point interpolation method (PIM), which possess the Kronecker delta property facilitating the straightforward enforcement of displacement boundary conditions. Using the generalized gradient smoothing technique, the “smoothed” strains at the middle points of the cells edges are first obtained using the corresponding edge-based smoothing domains and the assumed displacement field. In each triangular background cell, the strains at the vertices are assigned using these smoothed strains in a proper manner, and then piecewisely linear strain fields are constructed by the linear interpolation for each sub-triangular cell using the edge-based “smoothed” strains. With the assumed displacements and constructed linear strain fields, the discretized system equations are created using the Strain Constructed Galerkin (SC-Galerkin) weak form. A number of benchmark numerical examples, including the standard patch test, static, free and forced vibration problems, have been studied and intensive numerical results have demonstrated that the present method possesses the following properties: (1) it works well with the simplest triangular mesh, no additional degrees of freedom and parameters are introduced and very easy to implement; (2) it is at least linearly conforming; (3) it possesses a close-to-exact stiffness: it is much stiffer than the “overly-soft” node-based smoothed point interpolation method (NS-PIM) and much softer than the “overly-stiff” FEM model; (4) the results of the present method are of superconvergence and ultraaccuracy: about one order of magnitude more accurate than those of the linear FEM; (5) there are no spurious non-zeros energy modes found and it is also temporally stable, hence the present method works well for dynamic problems.